continuous
Let X and Y be topological spaces. A function f:X→Y is continuous
if, for every open set U⊂Y, the inverse image f-1(U) is an open subset of X.
In the case where X and Y are metric spaces (e.g. Euclidean space, or the space of real numbers), a function f:X→Y is continuous if and only if for every x∈X and every real number ϵ>0, there exists a real number δ>0 such that whenever a point z∈X has distance less than δ to x, the point f(z)∈Y has distance less than ϵ to f(x).
Continuity at a point
A related notion is that of local continuity, or continuity at a point (as opposed to the whole space X at once). When X and Y are topological spaces, we say f is continuous at a point x∈X if, for every open subset V⊂Y containing f(x), there is an open subset U⊂X containing x whose image f(U) is contained in V. Of course, the function f:X→Y is continuous in the first sense if and only if f is continuous at every point x∈X in the second sense (for students who haven’t seen this before, proving it is a worthwhile exercise).
In the common case where X and Y are metric spaces (e.g., Euclidean spaces), a function f is continuous at x∈X if and only if for every real number ϵ>0, there exists a real number δ>0 satisfying the property that dY(f(x),f(z))<ϵ for all z∈X with dX(x,z)<δ. Alternatively, the function f is continuous at a∈X if and only if the limit of f(x) as x→a satisfies the equation
lim |
Title | continuous |
Canonical name | Continuous |
Date of creation | 2013-03-22 11:51:55 |
Last modified on | 2013-03-22 11:51:55 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 12 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 26A15 |
Classification | msc 54C05 |
Classification | msc 81-00 |
Classification | msc 82-00 |
Classification | msc 83-00 |
Classification | msc 46L05 |
Synonym | continuous function |
Synonym | continuous map |
Synonym | continuous mapping |
Related topic | Limit |
Defines | continuous at |